theorem
  f is_simple_func_in S implies f|A is_simple_func_in S
proof
  assume
A1: f is_simple_func_in S;
  then Im f is_simple_func_in S by MESFUN7C:43;
  then R_EAL Im f is_simple_func_in S by MESFUNC6:49;
  then R_EAL((Im f)|A) is_simple_func_in S by MESFUNC5:34;
  then (Im f)|A is_simple_func_in S by MESFUNC6:49;
  then
A2: Im(f|A) is_simple_func_in S by MESFUN6C:7;
  Re f is_simple_func_in S by A1,MESFUN7C:43;
  then R_EAL Re f is_simple_func_in S by MESFUNC6:49;
  then R_EAL((Re f)|A) is_simple_func_in S by MESFUNC5:34;
  then (Re f)|A is_simple_func_in S by MESFUNC6:49;
  then Re(f|A) is_simple_func_in S by MESFUN6C:7;
  hence f|A is_simple_func_in S by A2,MESFUN7C:43;
end;
